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Mathlib.Topology.Category.Profinite.EffectiveEpi

Effective epimorphisms and finite effective epimorphic families in Profinite #

This file proves that Profinite is Preregular. Together with the fact that it is FinitaryPreExtensive, this implies that Profinite is Precoherent.

To do this, we need to characterise effective epimorphisms in Profinite. As a consequence, we also get a characterisation of finite effective epimorphic families.

Main results #

As a consequence, we obtain instances that Profinite is precoherent and preregular.

noncomputable def Profinite.struct {B : Profinite} {X : Profinite} (π : X B) (hπ : Function.Surjective π) :

Implementation: If π is a surjective morphism in Profinite, then it is an effective epi. The theorem Profinite.effectiveEpi_tfae should be used instead.

Equations
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Instances For
    theorem Profinite.effectiveEpiFamily_tfae {α : Type} [Finite α] {B : Profinite} (X : αProfinite) (π : (a : α) → X a B) :
    List.TFAE [CategoryTheory.EffectiveEpiFamily X π, CategoryTheory.Epi (CategoryTheory.Limits.Sigma.desc π), ∀ (b : B.toCompHaus.toTop), ∃ (a : α) (x : (X a).toCompHaus.toTop), (π a) x = b]
    theorem Profinite.effectiveEpiFamily_of_jointly_surjective {α : Type} [Finite α] {B : Profinite} (X : αProfinite) (π : (a : α) → X a B) (surj : ∀ (b : B.toCompHaus.toTop), ∃ (a : α) (x : (X a).toCompHaus.toTop), (π a) x = b) :